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The Kaczmarz and Gauss-Seidel methods aim to solve a linear $m \times n$ system $\boldsymbol{X} \boldsymbol{\beta} = \boldsymbol{y}$ by iteratively refining the solution estimate; the former uses random rows of $\boldsymbol{X}$ {to update $\boldsymbol{\beta}$ given the corresponding equations} and the latter uses random columns of $\boldsymbol{X}$ {to update corresponding coordinates in $\boldsymbol{\beta}$}.
Kaczmarz, S. [1937], ‘Angenäherte auflösung von systemen linearer gleichungen’, Bull. Int. Acad. Polon. Sci. Lett. Ser. A
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Reprint of the 1986 original. http://dx.doi.org/10.1137/1.9780898719284
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Ivanov, A. A. and Zhdanov, A. I. [2013], ‘Kaczmarz algorithm for tikhonov regularization problem’, Applied Mathematics E-Notes
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Needell, D. and Tropp, J. A. [2013], ‘Paved with good intentions: Analysis of a randomized block kaczmarz method’, Linear Algebra and its Applications
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Cited alongside, same era.
Popa, C., Preclik, T., Köstler, H. and Rüde, U. [2012], ‘On Kaczmarz’s projection iteration as a direct solver for linear least squares problems’, Linear Algebra and Its Applications
2012
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Richtárik, P. and Takáč, M. [2012 a
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Richtárik, P. and Takáč, M. [2012 b
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Later among the works it cites.
2013
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to appear
Needell, D., Sbrero, N. and Ward, R. [2014], ‘Stochastic gradient descent and the randomized kaczmarz algorithm’, Math. Program. Series A · 2014
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Ma, A., Needell, D. and Ramdas, A. [2015], ‘Convergence properties of the randomized extended Gauss-Seidel and Kaczmarz methods’, 36
2015
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